Finite Group Databasebeta

← #29577454

Subgroups — #29577454

Order 15616.

618 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 618 conjugacy classes of subgroups — too many to draw. Only its 115 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21448176C211yesmaximal, characteristic
2#21444339C212yesmaximal
3#21444149C211yesmaximal, characteristic
4#21443705C212yesmaximal
5C61×#2583C211yesmaximal, characteristic
6C2×#14519736—21no—
7#16839726C2212yes—
8C4⋊C4×Dic61—24no—
9C4×C22⋊Dic61—21no—
10C4×C22⋊Dic61C2212yes—
11#16839499—24no—
12#16839488—22no—
13C2×C22⋊Q8×C61C2211yescharacteristic
14C22×C4⋊C4×C61C2211yescharacteristic
15C61×#398C412yescomplemented
16C61×#395C412yescomplemented
17C61×#389—22no—
18C61×#385—21no—
19C61×#385—21no—
20C4×C22⋊C4×C61C2211yescharacteristic
21C2×#11758194—22no—
22#14519736—28no—
23C2×C4⋊Dic61—22no—
24C2×C22⋊Dic61—22no—
25C2×C4×Dic61—22no—
26C2×C4×Dic61—22no—
27C2×C4×Dic61—22no—
28C22⋊#92418—22no—
29C22⋊#92418—22no—
30#14519684—22no—
31#14519684D414yes—
32C22×Q8×C61C2×C411yescharacteristic
33C23×C4×C61D411yescharacteristic
34C23×C4×C61D411yescharacteristic
35C23×C4×C61C2311yescharacteristic
36C22⋊Q8×C61—22no—
37C22⋊Q8×C61—22no—
38C2×C4⋊C4×C61—22no—
39C2×C4⋊C4×C61—22no—
40C2×C4⋊C4×C61—22no—
41C2×C4⋊C4×C61—21no—
42C2×C4⋊C4×C61—22no—
43C2×C4⋊C4×C61C2×C411yescharacteristic
44C2×C4⋊C4×C61C2×C412yes—
45C2×C4⋊C4×C61D412yes—
46C2×C22⋊C4×C61—21no—
47C2×C22⋊C4×C61—21no—
48C2×C42×C61—21no—
49#95×C61—22no—
50#95×C61—22no—
51#95×C61C2×C412yes—
52C24×C61C2×D411yescharacteristic
53C2×Q8×C61—22no—
54C2×Q8×C61—81no—
55C22×C4×C61—21no—
56C22×C4×C61—21no—
57C22×C4×C61—24no—
58C22×C4×C61—22no—
59C22×C4×C61—21no—
60C22×C4×C61—21no—
61C22×C4×C61—21no—
62C22×C4×C61—21no—
63C22×C4×C61—81no—
64C22×C4×C61—81no—
65C22×C4×C61C22⋊C412yes—
66C22×C4×C61C22⋊C412yes—
67C22×C4×C61C22×C411yescharacteristic
68C22×C4×C61C2×D412yes—
69C22×C4×C61C2×D411yescharacteristic
70C22×C4×C61#4112yes—
71C22×Dic61—24no—
72C4⋊C4×C61—24no—
73C4⋊C4×C61—24no—
74C4⋊C4×C61—22no—
75C4⋊C4×C61—44no—
76C4⋊C4×C61—42no—
77C4⋊C4×C61—82no—
78C22⋊C4×C61—24no—
79C22⋊C4×C61—41no—
80C22⋊C4×C61—41no—
81C42×C61—81no—
82C22⋊Dic61—28no—
83C4⋊Dic61—28no—
84C4×Dic61—28no—
85C4×Dic61—82no—
86C4×Dic61—82no—
87#11758194—82no—
88C2×#92418—42no—
89C23×C61—22no—
90C23×C61—22no—
91C23×C61—41no—
92C23×C61C2×C22⋊C411yescharacteristic
93C23×C61#12011yescharacteristic
94C23×C61#12311yescharacteristic
95Q8×C61—82no—
96C2×C4×C61—22no—
97C2×C4×C61—22no—
98C2×C4×C61—24no—
99C2×C4×C61—22no—
100C2×C4×C61—22no—
101C2×C4×C61—24no—
102C2×C4×C61—22no—
103C2×C4×C61—42no—
104C2×C4×C61—42no—
105C2×C4×C61—41no—
106C2×C4×C61—41no—
107C2×C4×C61—42no—
108C2×C4×C61—42no—
109C2×C4×C61—42no—
110C2×C4×C61—42no—
111C2×C4×C61—42no—
112C2×C4×C61—41no—
113C2×C4×C61—41no—
114C2×C4×C61—42no—
115C2×C4×C61—81no—
116C2×C4×C61#10414yes—
117C2×C4×C61C4×D412yes—
118C2×C4×C61#12112yes—
119C2×Dic61—28no—
120C2×Dic61—44no—
121C2×Dic61—44no—
122#92418—82no—
123#9408—611nomaximal, Sylow
124C22×C61—21no—
125C22×C61—21no—
126C22×C61—21no—
127C22×C61—21no—
128C22×C61—21no—
129C22×C61—21no—
130C22×C61—21no—
131C22×C61—44no—
132C22×C61#38611yescharacteristic
133C22×C61C2×#10412yes—
134C22×C61#45612yes—
135C4×C61—24no—
136C4×C61—22no—
137C4×C61—22no—
138C4×C61—44no—
139C4×C61—42no—
140C4×C61—81no—
141C4×C61—81no—
142Dic61—48no—
143#2583D6111yescharacteristic
144#2274—611no—
145#1508—612no—
146#1463—611no—
147#1297—612no—
148C2×C61—22no—
149C2×C61—22no—
150C2×C61—41no—
151C2×C61#188611yescharacteristic
152C2×C61#189611yescharacteristic
153C2×C61#191311yescharacteristic
154C2×C22⋊Q8C2×D6111yescharacteristic
155C2×#117—1221no—
156C22×C4⋊C4C2×D6111yescharacteristic
157#407—612no—
158#398Dic6112yescomplemented
159#395Dic6112yescomplemented
160#389—22no—
161#385—21no—
162#385—21no—
163C4×C4⋊C4—1224no—
164C4×C22⋊C4—612no—
165C4×C22⋊C4—1221no—
166C4×C22⋊C4C2×D6111yescharacteristic
167#328—1224no—
168#324—1222no—
169C61#940811yescharacteristic, complemented, Sylow
170C22×Q8C2×Dic6111yescharacteristic
171C23×C4C22⋊D6111yescharacteristic
172C23×C4C22⋊D6111yescharacteristic
173C23×C4C22×D6111yescharacteristic
174C2×M16—1222no—
175C22⋊Q8—22no—
176C22⋊Q8—22no—
177#117—1228no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—1222no—
184C2×C4⋊C4C2×Dic6111yescharacteristic
185C2×C4⋊C4C2×Dic6112yes—
186C2×C4⋊C4C22⋊D6112yes—
187C2×C22⋊C4—21no—
188C2×C22⋊C4—21no—
189C2×C22⋊C4—1222no—
190C2×C42—21no—
191C2×C42—1222no—
192C2×C42—1222no—
193C2×C42—1222no—
194C22⋊C8—1222no—
195C22⋊C8—1222no—
196#95—22no—
197#95—22no—
198#95—614no—
199#95—1222no—
200#95C2×Dic6112yes—
201C24C2×C22⋊D6111yescharacteristic
202C2×Q8—22no—
203C2×Q8—81no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—24no—
207C22×C4—22no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—81no—
213C22×C4—81no—
214C22×C4—1224no—
215C22×C4C22⋊Dic6112yes—
216C22×C4C22⋊Dic6112yes—
217C22×C4D4×D6112yes—
218C22×C4#1175822412yes—
219C22×C4C22×Dic6111yescharacteristic
220C22×C4C2×C22⋊D6111yescharacteristic
221M16—4882no—
222C2×C8—2442no—
223C4⋊C4—24no—
224C4⋊C4—24no—
225C4⋊C4—22no—
226C4⋊C4—44no—
227C4⋊C4—42no—
228C4⋊C4—82no—
229C4⋊C4—1228no—
230C22⋊C4—24no—
231C22⋊C4—41no—
232C22⋊C4—41no—
233C22⋊C4—1228no—
234C42—81no—
235C42—1228no—
236C42—4882no—
237C42—4882no—
238C23—22no—
239C23—22no—
240C23—41no—
241C23C2×C22⋊Dic6111yescharacteristic
242C23#1451974511yescharacteristic
243C23#1451975811yescharacteristic
244Q8—82no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—24no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—24no—
251C2×C4—22no—
252C2×C4—41no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—41no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—81no—
265C2×C4—1228no—
266C2×C4—2444no—
267C2×C4—2444no—
268C2×C4#1451970414yes—
269C2×C4D4×Dic6112yes—
270C2×C4#1451974912yes—
271C8—4882no—
272C22—21no—
273C22—21no—
274C22—21no—
275C22—21no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22—44no—
280C22#1683964211yescharacteristic
281C22C2×#1451970412yes—
282C22#1683994412yes—
283C4—24no—
284C4—22no—
285C4—22no—
286C4—44no—
287C4—42no—
288C4—81no—
289C4—81no—
290C4—2448no—
291C2—22no—
292C2—22no—
293C2—41no—
294C2#2144621311yescharacteristic
295C2#2144626311yescharacteristic
296C2#2144636211yescharacteristic
subgroupG/H
centerC22#16839642
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#2583C2
socleC22×C61#386
prime radicalC24×C61C2×D4
cosocleC23×C4×C61C23
hypercenter#2583D61
hypocenterC61#9408
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2583

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
15. C61×#398C4semidirect
16. C61×#395C4semidirect
158. #398Dic61semidirect
159. #395Dic61semidirect
169. C61#9408semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#214481762
#214443392
#214441492
#214437052
C61×#25832
#940861